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| Mirrors > Home > MPE Home > Th. List > Mathboxes > madjusmdet | Structured version Visualization version GIF version | ||
| Description: Express the cofactor of the matrix, i.e. the entries of its adjunct matrix, using determinant of submatrices. (Contributed by Thierry Arnoux, 23-Aug-2020.) | 
| Ref | Expression | 
|---|---|
| madjusmdet.b | ⊢ 𝐵 = (Base‘𝐴) | 
| madjusmdet.a | ⊢ 𝐴 = ((1...𝑁) Mat 𝑅) | 
| madjusmdet.d | ⊢ 𝐷 = ((1...𝑁) maDet 𝑅) | 
| madjusmdet.k | ⊢ 𝐾 = ((1...𝑁) maAdju 𝑅) | 
| madjusmdet.t | ⊢ · = (.r‘𝑅) | 
| madjusmdet.z | ⊢ 𝑍 = (ℤRHom‘𝑅) | 
| madjusmdet.e | ⊢ 𝐸 = ((1...(𝑁 − 1)) maDet 𝑅) | 
| madjusmdet.n | ⊢ (𝜑 → 𝑁 ∈ ℕ) | 
| madjusmdet.r | ⊢ (𝜑 → 𝑅 ∈ CRing) | 
| madjusmdet.i | ⊢ (𝜑 → 𝐼 ∈ (1...𝑁)) | 
| madjusmdet.j | ⊢ (𝜑 → 𝐽 ∈ (1...𝑁)) | 
| madjusmdet.m | ⊢ (𝜑 → 𝑀 ∈ 𝐵) | 
| Ref | Expression | 
|---|---|
| madjusmdet | ⊢ (𝜑 → (𝐽(𝐾‘𝑀)𝐼) = ((𝑍‘(-1↑(𝐼 + 𝐽))) · (𝐸‘(𝐼(subMat1‘𝑀)𝐽)))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | madjusmdet.b | . 2 ⊢ 𝐵 = (Base‘𝐴) | |
| 2 | madjusmdet.a | . 2 ⊢ 𝐴 = ((1...𝑁) Mat 𝑅) | |
| 3 | madjusmdet.d | . 2 ⊢ 𝐷 = ((1...𝑁) maDet 𝑅) | |
| 4 | madjusmdet.k | . 2 ⊢ 𝐾 = ((1...𝑁) maAdju 𝑅) | |
| 5 | madjusmdet.t | . 2 ⊢ · = (.r‘𝑅) | |
| 6 | madjusmdet.z | . 2 ⊢ 𝑍 = (ℤRHom‘𝑅) | |
| 7 | madjusmdet.e | . 2 ⊢ 𝐸 = ((1...(𝑁 − 1)) maDet 𝑅) | |
| 8 | madjusmdet.n | . 2 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
| 9 | madjusmdet.r | . 2 ⊢ (𝜑 → 𝑅 ∈ CRing) | |
| 10 | madjusmdet.i | . 2 ⊢ (𝜑 → 𝐼 ∈ (1...𝑁)) | |
| 11 | madjusmdet.j | . 2 ⊢ (𝜑 → 𝐽 ∈ (1...𝑁)) | |
| 12 | madjusmdet.m | . 2 ⊢ (𝜑 → 𝑀 ∈ 𝐵) | |
| 13 | eqeq1 2740 | . . . 4 ⊢ (𝑘 = 𝑖 → (𝑘 = 1 ↔ 𝑖 = 1)) | |
| 14 | breq1 5145 | . . . . 5 ⊢ (𝑘 = 𝑖 → (𝑘 ≤ 𝐼 ↔ 𝑖 ≤ 𝐼)) | |
| 15 | oveq1 7439 | . . . . 5 ⊢ (𝑘 = 𝑖 → (𝑘 − 1) = (𝑖 − 1)) | |
| 16 | id 22 | . . . . 5 ⊢ (𝑘 = 𝑖 → 𝑘 = 𝑖) | |
| 17 | 14, 15, 16 | ifbieq12d 4553 | . . . 4 ⊢ (𝑘 = 𝑖 → if(𝑘 ≤ 𝐼, (𝑘 − 1), 𝑘) = if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖)) | 
| 18 | 13, 17 | ifbieq2d 4551 | . . 3 ⊢ (𝑘 = 𝑖 → if(𝑘 = 1, 𝐼, if(𝑘 ≤ 𝐼, (𝑘 − 1), 𝑘)) = if(𝑖 = 1, 𝐼, if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖))) | 
| 19 | 18 | cbvmptv 5254 | . 2 ⊢ (𝑘 ∈ (1...𝑁) ↦ if(𝑘 = 1, 𝐼, if(𝑘 ≤ 𝐼, (𝑘 − 1), 𝑘))) = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝐼, if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖))) | 
| 20 | breq1 5145 | . . . . 5 ⊢ (𝑘 = 𝑖 → (𝑘 ≤ 𝑁 ↔ 𝑖 ≤ 𝑁)) | |
| 21 | 20, 15, 16 | ifbieq12d 4553 | . . . 4 ⊢ (𝑘 = 𝑖 → if(𝑘 ≤ 𝑁, (𝑘 − 1), 𝑘) = if(𝑖 ≤ 𝑁, (𝑖 − 1), 𝑖)) | 
| 22 | 13, 21 | ifbieq2d 4551 | . . 3 ⊢ (𝑘 = 𝑖 → if(𝑘 = 1, 𝑁, if(𝑘 ≤ 𝑁, (𝑘 − 1), 𝑘)) = if(𝑖 = 1, 𝑁, if(𝑖 ≤ 𝑁, (𝑖 − 1), 𝑖))) | 
| 23 | 22 | cbvmptv 5254 | . 2 ⊢ (𝑘 ∈ (1...𝑁) ↦ if(𝑘 = 1, 𝑁, if(𝑘 ≤ 𝑁, (𝑘 − 1), 𝑘))) = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝑁, if(𝑖 ≤ 𝑁, (𝑖 − 1), 𝑖))) | 
| 24 | eqeq1 2740 | . . . 4 ⊢ (𝑙 = 𝑗 → (𝑙 = 1 ↔ 𝑗 = 1)) | |
| 25 | breq1 5145 | . . . . 5 ⊢ (𝑙 = 𝑗 → (𝑙 ≤ 𝐽 ↔ 𝑗 ≤ 𝐽)) | |
| 26 | oveq1 7439 | . . . . 5 ⊢ (𝑙 = 𝑗 → (𝑙 − 1) = (𝑗 − 1)) | |
| 27 | id 22 | . . . . 5 ⊢ (𝑙 = 𝑗 → 𝑙 = 𝑗) | |
| 28 | 25, 26, 27 | ifbieq12d 4553 | . . . 4 ⊢ (𝑙 = 𝑗 → if(𝑙 ≤ 𝐽, (𝑙 − 1), 𝑙) = if(𝑗 ≤ 𝐽, (𝑗 − 1), 𝑗)) | 
| 29 | 24, 28 | ifbieq2d 4551 | . . 3 ⊢ (𝑙 = 𝑗 → if(𝑙 = 1, 𝐽, if(𝑙 ≤ 𝐽, (𝑙 − 1), 𝑙)) = if(𝑗 = 1, 𝐽, if(𝑗 ≤ 𝐽, (𝑗 − 1), 𝑗))) | 
| 30 | 29 | cbvmptv 5254 | . 2 ⊢ (𝑙 ∈ (1...𝑁) ↦ if(𝑙 = 1, 𝐽, if(𝑙 ≤ 𝐽, (𝑙 − 1), 𝑙))) = (𝑗 ∈ (1...𝑁) ↦ if(𝑗 = 1, 𝐽, if(𝑗 ≤ 𝐽, (𝑗 − 1), 𝑗))) | 
| 31 | breq1 5145 | . . . . 5 ⊢ (𝑙 = 𝑗 → (𝑙 ≤ 𝑁 ↔ 𝑗 ≤ 𝑁)) | |
| 32 | 31, 26, 27 | ifbieq12d 4553 | . . . 4 ⊢ (𝑙 = 𝑗 → if(𝑙 ≤ 𝑁, (𝑙 − 1), 𝑙) = if(𝑗 ≤ 𝑁, (𝑗 − 1), 𝑗)) | 
| 33 | 24, 32 | ifbieq2d 4551 | . . 3 ⊢ (𝑙 = 𝑗 → if(𝑙 = 1, 𝑁, if(𝑙 ≤ 𝑁, (𝑙 − 1), 𝑙)) = if(𝑗 = 1, 𝑁, if(𝑗 ≤ 𝑁, (𝑗 − 1), 𝑗))) | 
| 34 | 33 | cbvmptv 5254 | . 2 ⊢ (𝑙 ∈ (1...𝑁) ↦ if(𝑙 = 1, 𝑁, if(𝑙 ≤ 𝑁, (𝑙 − 1), 𝑙))) = (𝑗 ∈ (1...𝑁) ↦ if(𝑗 = 1, 𝑁, if(𝑗 ≤ 𝑁, (𝑗 − 1), 𝑗))) | 
| 35 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 19, 23, 30, 34 | madjusmdetlem4 33830 | 1 ⊢ (𝜑 → (𝐽(𝐾‘𝑀)𝐼) = ((𝑍‘(-1↑(𝐼 + 𝐽))) · (𝐸‘(𝐼(subMat1‘𝑀)𝐽)))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 ifcif 4524 class class class wbr 5142 ↦ cmpt 5224 ‘cfv 6560 (class class class)co 7432 1c1 11157 + caddc 11159 ≤ cle 11297 − cmin 11493 -cneg 11494 ℕcn 12267 ...cfz 13548 ↑cexp 14103 Basecbs 17248 .rcmulr 17299 CRingccrg 20232 ℤRHomczrh 21511 Mat cmat 22412 maDet cmdat 22591 maAdju cmadu 22639 subMat1csmat 33793 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-addf 11235 ax-mulf 11236 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-xor 1511 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-ot 4634 df-uni 4907 df-int 4946 df-iun 4992 df-iin 4993 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-se 5637 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-of 7698 df-om 7889 df-1st 8015 df-2nd 8016 df-supp 8187 df-tpos 8252 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-2o 8508 df-er 8746 df-map 8869 df-pm 8870 df-ixp 8939 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-fsupp 9403 df-sup 9483 df-oi 9551 df-card 9980 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-xnn0 12602 df-z 12616 df-dec 12736 df-uz 12880 df-rp 13036 df-fz 13549 df-fzo 13696 df-seq 14044 df-exp 14104 df-hash 14371 df-word 14554 df-lsw 14602 df-concat 14610 df-s1 14635 df-substr 14680 df-pfx 14710 df-splice 14789 df-reverse 14798 df-s2 14888 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-0g 17487 df-gsum 17488 df-prds 17493 df-pws 17495 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-mhm 18797 df-submnd 18798 df-efmnd 18883 df-grp 18955 df-minusg 18956 df-mulg 19087 df-subg 19142 df-ghm 19232 df-gim 19278 df-cntz 19336 df-oppg 19365 df-symg 19388 df-pmtr 19461 df-psgn 19510 df-cmn 19801 df-abl 19802 df-mgp 20139 df-rng 20151 df-ur 20180 df-ring 20233 df-cring 20234 df-oppr 20335 df-dvdsr 20358 df-unit 20359 df-invr 20389 df-dvr 20402 df-rhm 20473 df-subrng 20547 df-subrg 20571 df-drng 20732 df-sra 21173 df-rgmod 21174 df-cnfld 21366 df-zring 21459 df-zrh 21515 df-dsmm 21753 df-frlm 21768 df-mat 22413 df-marrep 22565 df-subma 22584 df-mdet 22592 df-madu 22641 df-minmar1 22642 df-smat 33794 | 
| This theorem is referenced by: mdetlap 33832 | 
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